Comprehensive grouping efficacy: A new measure for evaluating block-diagonal forms in group technology

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1 International Journal of Industrial Engineering Comutations 9 (08) 55 7 Contents lists available at GrowingScience International Journal of Industrial Engineering Comutations homeage: Comrehensive grouing efficacy: A new measure for evaluating bloc-diagonal forms in grou technology Adnan Muattash a*, Nadia Dahmani a,b, Adnan Al-ashir c and Ahmad Qamar c a Deartment of Industrial Management, Emirates College of Technology, Abu Dhabi, UAE b LARODEC Laboratory, Institut Suerieur de Gestion, 000 Le ardo, Tunisia c Deartment of Industrial Engineering, Faculty of Engineering, Hashemite University, Zara Jordan C H R O N I C L E A S T R A C T Article history: Received October 7 06 Received in Revised Format December 5 06 Acceted March 07 Available online March 4 07 Keywords: Cell Size Grouing measures Sarsity Sarsity index Comrehensive Grouing measure Efficiency index The goodness of machine-art grous in cellular manufacturing systems is evaluated by different measures available in the literature. The commonly nown grouing efficiency measures will be discussed in this aer. None of these measures has the ability to evaluate the efficiency of bloc -diagonal system and sub-system at the same time. Moreover, sarsity of individual cells was not taen into consideration in these measures. In this aer, a new grouing measure called Comrehensive Grouing Efficacy (CGE) is roosed to overcome the drawbacs of these measures. CGE is tested against some roblems from the literature and the results demonstrate the ability of this measure to be used as comrehensive grouing measure since four of the wellnown measures are included in the CGE formula. The sueriority of CGE is that it can be used to find the efficiency of bloc-diagonal form, the efficiency of sub-system, sarsity index and efficacy index at the same time, which will give the designer the oortunity to control the cell size. Without nowing the efficiency of sub-systems (individual cells), the system designer will not be able to control the cell size. 08 Growing Science Ltd. All rights reserved. Introduction Grou technology (GT) is a method of organizing and using information about comonent similarities to imrove the roduction efficiency of small to medium batch oriented manufacturing systems (Asin & Chiu, 990). The main idea of GT is to caitalize on similar manufacturing rocesses and features where similar arts are groued into a art family and manufactured by a cluster of dissimilar machines (Wu, 998). The inut to the GT roblem is a zero-one matrix A where a ij = indicates the visit of comonent j to machine i, and a ij = 0 otherwise. Grouing of comonents into families and machines into cells results in a transformed matrix with diagonal-blocs where ones occuy the diagonal-blocs and zeros occuy the off-diagonal blocs. The resulting diagonal blocs reresent the manufacturing cells. Cellular manufacturing (CM) is an imortant alication of grou technology (GT) in which sets * Corresonding author adnan.muqatash@ect.ac.ae (A. Muattash) 07 Growing Science Ltd. All rights reserved. doi: 0.567/j.ijiec

2 56 (families) of arts are roduced on a grou of various machines, which are hysicaly close together and can entirely rocess a family of arts. The identification of art families and machine grous in the design of cellular manufacturing systems is commonly referred to as cell design/formation (Mansouri et al., 000). Algorithms that aim at forming the art families and machine cells essentially try to rearrange the rows and columns of art/machine incidence matrix to get a bloc-diagonal form. Different methods are available in the literature (Elbenani, & Ferland, 0; rusco, 05; ychov, et al., 04; Ghosh et al., 04; Murugan & Selladurai 0; ottani et al., 07; Rezazadeh & Khiali-Miab, 07; Rabbani et al., 07). The size of each cell, measured by the number of machines allocated to the cell, is a variable that needs to be controlled. There are several reasons (e.g. available sace and visible control requirements) that might imose an uer limit on the number of machines. Also, it is not usual to construct a cell of one or two machines. This may lead to a very low utilization of the cell s handling and loading equiment. That is why there must be an uer and lower bounds on cell size (octor, 996). The ideal situation is the one in which all the ones are in the diagonal-blocs and all the zeros are in the off-diagonal blocs. However, the ideal case seldom occurs in for a real sho floor roblem (Kumar & Chandraseharom, 990). The structure of the final machine-comonent matrix significantly affects the effectiveness of the corresonding cellular manufacturing system (Seifoddini & Djassem, 996). For this reason the choice of grouing methodology must be based on criteria that can indicate the goodness of a grouing solution. Hence, a number of grouing measures have been develoed to evaluate the efficiency of bloc-diagonal forms. Some of these measures are, Grouing caability index (GCI) (Hsu, 990), Global efficiency (GLE) (Harhalais et al., 990), Grouing measure (Miltenburg & Zhang, 99), Weighted Grouing Efficiency (Sarar & Khan, 00) and Double weighted grouing efficiency (Sarar, 00), GT efficacy (Kichun & Ahn, 03), Modified grouing efficacy (Rajesh et al., 06). Some other well-nown measures will be discussed below in section. For other measures that are available in the literature see (Sarer & Mondal, 999; Sarer & Khan, 00, Sarer, 00; Keeling et al., 007, Agrawal et al., 0, Kichun & Kwang-Il, 03). Kichun Lee and Kwang-Il Ahn (03) ointed out that, grouing efficacy is used as a standard measure for evaluating solutions based on a binary art-machine matrix. None of the above mentioned measures can evaluate the efficiency of bloc-diagonal system and subsystem at the same time which means that, these measures do not have the ability to determine (or quantify) the quality of individual cells inside the matrix. Moreover, sarsity of individual cells was not taen into consideration in these measures and none of these measures can rovide the system designer with any cell indicator that can hel him to control the cell size. This aer introduces a new measure called Comrehensive Grouing Efficacy (CGE) which is considered to be more accurate to determine the efficiency of a bloc-diagonal form for develoing cellular manufacturing systems. The main features of CGE measure are: First, CGE can find the efficiency of bloc-diagonal system and, at the same time, it can reflect the goodness of every cell by taing into consideration the number of oerations, number of voids, number of excetional arts, cell size (sarsity of individual cell in the solved matrix) and sarsity of the system regardless of the size of the matrix. Second, CGE is a comrehensive grouing measure since it can be used to find the efficiency of bloc-diagonal system and/or cell utilization and/or machine utilization and/or cell indicator and/or cell flexibility at the same time. Finally, CGE will rovide the designer with three indicators (sarsity index, efficacy index and efficiency of individual cells in the solved matrix) to control the cell size. The following definitions will be used in this aer: loc: A sub-matrix of the machine comonent incidence matrix formed by the intersection of columns reresenting a comonent family and rows reresenting a machine cell. Voids: A zero element aearing in a diagonal bloc. Excetional element (or excetion): The one aearing in the off-diagonal blocs. Perfect bloc-diagonal form: The bloc-diagonal form in which all diagonal blocs contain ones and all off-diagonal blocs contain zeros (Kumar & Chandrasehoran, 990).

3 A. Muattash et al. / International Journal of Industrial Engineering Comutations 9 (08) 57 Sarsity (loc- diagonal sace): Total number of elements within the diagonal blocs of the solved matrix (Sarer & Khan, 00).. Literature Review The commonly nown grouing efficiency measures in the literature can be classified into two grous based on the efficiency evaluation of bloc-diagonal forms and evaluation of individual cells.. Efficiency evaluation of bloc-diagonal forms These measures are develoed to evaluate the efficiency of bloc-diagonal forms. Some of these measures are listed below. Grouing Efficiency (η): (Chandrasehoran & Rajogoalan, 986) The main drawbacs of GE have been exosed already in earlier studies (for more details see Kumar and Chandraseharan (990), Sarer and Mondal (999), Sarer and Khan (00) and Sarer (00)). It is defined as: Grouing Efficiency= q ( q) () ed where MrNr r and r e o M N r r. e d=total number of oerations in the Machine Part (MP) matrix, e 0=number of excetions, e v=number of voids, q=weighted factor, 0 q m= total number of arts in the matrix, n = total number of machines in the matrix. Machine Utilization (MU): (Chandraseharan & Rajagoalan, 986) The main drawbacs of MU are that number of voids and excetions are not taen into consideration. N MU c mn where c c c, () N : total number of,s in the diagonal blocs of the machine-art incident matrix, nc:total number of arts in the cth cell, m :total number of machines in the cth cell. c Grouing Efficacy (): (Suresh Kumar & Chandraseharan, 990) To overcome the roblems of (η ), grouing efficacy has been introduced. The most used measure in the literature is the Grouing efficacy. The main drawbac of ( ) is that, sarsity of the individual cells size is not taen into consideration. Grouing efficacy () is defined as:

4 58 (3), where Number of excetional elements and Total number of oerations in the MP matrix Number of voids in the diagonal blocs. Total number of oerations in the MP marix v e 0 (4) +e: total number of oerations in the MP matrix, : number of oerations in the diagonal bloc, e: number of excetions, v: number of voids. Weighted grouing efficacy ( ): (Ng, 993) The main drawbac of ( ) is that, sarsity of the individual cells size is not taen into consideration qe ( e0 ) qe ( e e0) ( qe ) 0 v where e: total number of oerations in the MP matrix, e 0: number of excetions, e v: number of voids, q:weighted factor. (5) Grouing Index (γ): (Nair & Narendran, 996) is derived from the modified grouing efficacy by introducing a correction factor. The main drawbac of ( ) is that, sarsity of the individual cells in the solved matrix is not taen into consideration qev ( q)( e0 A), qev ( q)( e0 A) where A 0 for e0 and A e0- for e0 greater than can be written as follows, (6) qev ( - q)( e0 - A) qev ( - q)( e0 - A) -,where and -,where and A is a correction factor and is the sarsity of the solved matrix and e 0 is the number of excetions, e v is the number of voids and q is the weighted factor. Modified grouing efficacy ( ): (Nair & Narendran, 996) The main drawbac of this measure is that, sarsity of the individual cells in the solved matrix is not taen into consideration. qev ( q) e0 (7), qe ( q) e0 v where is the sarsity of the solved matrix, e0, ev and q reresent, the number of voids, the weighted factor and the number of excetions, resectively.

5 A. Muattash et al. / International Journal of Industrial Engineering Comutations 9 (08) 59. Evaluation of the individual cells in the solved matrix These measures are develoed to evaluate the individual cells in the solved matrix. Some of these measures are listed below. Cell Utilization (CU): (Mahdavi et al., 007) CU does not tae into consideration the excetional elements of the formed cells. This measure is used only to determine the utilization of individual cells inside the solved matrix. Cell utilization is defined as a number of non zero elements of bloc-diagonal divided by bloc-diagonal matrix size of each cell. Cell Utilization can be written as: CU Number of Ooerations in cell, loc-diagonal Matrix Size of cell (8) CU doesn t tae into consideration the excetional elements of the formed cells. Cell Indicator (α): (Al-ashir et al., 06) This measure is used only to determine the cell indicator of individual cells inside the solved matrix. The effect of individual cell size is not taen into consideration in this measure. v e, where α, v, e and reresent cell indicator of the th cell, the number of voids in th diagonal bloc, the number of excetional elements in the th off-diagonal bloc and the number of oerations in the th diagonal bloc, resectively. Measure of Flexibility (MF): (Nagendra, 004c) This measure is used only to determine the average measure of flexibility of individual cells inside the solved matrix. The effect of individual cell size is not taen into consideration in this measure. NO MF, nc ( m c ) where NO, m, c and nc reresent the number of oerations executed in the th cell, the number of machines in the th cell, the number of comonents in the th cell and the number of cells, resectively. Since MF varies from cell to cell, for evaluation uroses, the average measure of the flexibility is comuted as follows, MF () AMF 00, n c where nc is the number of cells formed. Obviously, higher values of AFM reresents higher flexibility. (9) (0) Illustration : Consider the solution matrix of Table (Al-asher et al., 06), which contains twelve machines and twelve arts. This case study will be used to clarify the difference between the two grous of the evaluation measures of both bloc-diagonal forms and individual cells (Eqs. -).

6 60 Table Final solution for illustration Machines Assume that the system designer wants to find the efficiency of the system shown in Table ; in this case one of the formulas shown in Eqs. (-7) will be used. Suose that grouing efficacy measure is used (Eq. 3), the result will be as shown in Table. Moreover, if the designer wants to evaluate the individual cells, in this case he/she will choose one of the formulas shown in Eqs. (8-). Assume that cell utilization is used (Eq. 8), the results are shown in Table. We can conclude that, the designer has to use two different formulas from the above two grous to find the efficiency of bloc -diagonal system and to evaluate the individual cells. Table Cell utilization and grouing efficacy roblem Table Cell utilization Cell Cell Solution Cell Grouing Efficacy ( ) 0.64 To avoid using two different formulas, a new Comrehensive Grouing Efficacy (CGE) measure is develoed in this aer shown in Eq. (3). CGE measure can be:. Rewritten in four different forms to : Find the efficiency of bloc-diagonal system and cell utilization at the same time. In this case CGE (Eq. 3) will be rewritten to include Eq. (8) inside the CGE formula as shown in Eq. (5). Find the efficiency of bloc-diagonal system and cell indicator at the same time. In this case CGE (Eq. 3) will be rewritten to include Eq. (9) inside the CGE formula as shown in Eq. (7). Find the efficiency of bloc-diagonal system and machine utilization at the same time. In this case CGE (Eq. 3) will be rewritten to include Eq. () inside the CGE formula as shown in Eq. (9). Find the efficiency of bloc-diagonal system and cell flexibility at the same time. In this case CGE (Eq. 3) will be rewritten to include Eq. (0) inside the CGE formula as shown in Eq. ().. Used as any other grouing measure to find the efficiency of bloc-diagonal system (Eq. 3). The sueriority of CGE is that the efficiency of bloc-diagonal system, the efficiency of sub-system, sarsity index and efficacy index can be found at the same time as shown in Eqs. (3-4). Without nowing the efficiency of sub-systems (individual cells), the system designer will not be able to control the cell size..3. Comarative study of different grouing measures In this section, we comare between the grouings measures mentioned in section through a case study from the literature and analyze their corresonding results.

7 A. Muattash et al. / International Journal of Industrial Engineering Comutations 9 (08) 6 Problem : Consider the solution matrix of Table 3, 4 and Table 5, taen from the literature (Kusia & Chow, 987). Table 3 Final solution matrix X for roblem Problem X Table 4 Final solution matrix Y for roblem Problem Y Table 5 Final solution matrix Z for roblem Problem Z Alying the different measures of goodness discussed earlier to evaluate the quality of the above different solutions, the results are obtained and summarized in Table 6 and Table 7.

8 6 Table 6 Evaluation of different measures for roblem (efficiency of bloc-diagonal form)(q=0.5) e+v η Table # machines in st cell # machines in nd cell # machines in 3 rd cell # arts in st cell # arts in nd cell # arts in 3 rd cell MU From Table 6, it is clear that there is a big difference between the grouing measures in the final results. In revious studies, it is considered that grouing efficacy is considered to be the most used measure. None of these measures taes into account the efficiency of individual cells inside the system. Cell size is not taen into considerations on these measures, for such reasons these measures do not reflect the efficiency of individual cells in truthful way. In other words the sarsity of individual cells in the solved matrix is not taen into considerations. Table 7 Evaluation of different measures for roblem (evaluation of individual cells) Table # machines in st cell # machines in nd cell # machines in 3 rd cell # arts in st cell # arts in nd cell # arts in 3 rd cell e+v Cell Utilization (CU) Cell Indicator (α) Average Measure of Flexibility (AMF) CU CU CU3 CU CU CU3 CU CU CU % 5% 5% % 9.8% 6.% % 9.06% 4.33% Table 7 shows different measures to find the utilization, efficiency and flexibility of the cells inside the solved matrix. None of these measures reflected the efficiency of bloc-diagonal form. Moreover, none of them either in Table 6 or Table 7 could find both system efficiency and evaluate the individual cells in one formula. 3. Proosed Measure 3. Comrehensive Grouing Efficacy In this section, a new grouing measure called Comrehensive Grouing Efficacy (CGE) is roosed to overcome these limitations. The new grouing measure can be exressed as: CGE CGE j j j j vj ej... ve v e v e () (3) where,, and reresent the sarsity of the first, the second and the th cell in the solved matrix, resectively. Also, reresents the sarsity of the solved matrix, which is defined as the total number of elements within diagonal blocs of the solve matrix. Here reresents the sarsity of the solved matrix n m n m and n m. Let α =/, α =/ and α =/ reresent the and,

9 A. Muattash et al. / International Journal of Industrial Engineering Comutations 9 (08) 63 sarsity index of the first, the second and the th cells, resectively. Let v e and v e, v e reresent the efficacy index of the first, the second and th cell, resectively with CGE..., (4) where α τ, α τ and α τ reresent the efficiency of the first, the second and he th cell, resectively. Here we have, m = total number of arts in the matrix, n = total number of machines in the matrix, m = number of arts in the jth diagonal bloc [jth cell], n = number of machines in the jth diagonal bloc [jth cell], v = number of voids in the jth diagonal bloc, e = number of excetional elements in the jth off-diagonal bloc, = number of oerations in the jth diagonal bloc, = total number of diagonal blocs [total number of cells in the matrix]. From the definition of CGE, it is clear that this new measure reflects the goodness of every cell by taing into consideration the number of oerations, number of voids, number of excetional arts, cell size (sarsity of individual cell in the solved matrix) and sarsity of the system regardless of the size of the matrix. Since CGE is the sum of efficiency of all individual cells, then the designer can discover which cell has the smallest efficiency, which will hel him to control the cell size. 3. Mathematical Proerties of Comrehensive Grouing Efficacy function. Non negativity: All the elements of comrehensive grouing measure are ositive.. Physical meaning of extremes: a. When all the ones in the erfect diagonal- bloc are outside the diagonal bloc [condition of zero efficiency], then CGE= 0 because. = 0. b. For erfect diagonal bloc [condition of 00% efficiency], then CGE = because v = v = v = 0, and e = e = e =0 and = , then CGE= c. From roerty and roerty it is found that 0CGE. 3.3 Sueriority of the Comrehensive Grouing Efficacy In this section, we highlight the merits of CGE comaring to the other measures.. CGE measure (Eq.3) can be used to find the efficiency of bloc-diagonal form, the efficiency of sub-system, sarsity index and efficacy index at the same time. Any one of these three indicators (efficiency of sub-system, sarsity index and efficacy index) will give the system designer the oortunity to control the cell size. Comrehensive Grouing Efficacy measure (CGE) CGE measure (Eq. 3) can be used as any other grouing efficiency measure to evaluate bloc-diagonal forms in grou technology. Sarsity Index ( ) It is defined as the ratio of the sarsity of sub-system to the system sarsity. The imortance of sarsity index is that it reflects the imact of every cell size to the sarsity

10 64 of the solved matrix, which will hel the designer control the cell size through reformation of manufacturing systems, machines allocation or art assignment. As the sarsity index increases, the efficiency of individual cell will increase. Efficacy Index ( ) It reflects the number of oerations for cell j to the size of cell j and the number of excetions belongs to cell j. Knowing efficacy index will hel the designer now the voids, excetions and number of oerations in each cell. Then cell size can be controlled through art assignment. Part assignment is erformed to minimize the number of voids inside the cells and number of ones outside the cells. This aroach gives the system designer the ability to control the lower and/or the uer bound of cell size. Efficiency of sub-system (individual cells in the solved matrix). It is defined as Sarsity Index ( ) multilied by Efficacy Index ( ). The efficiency of individual cell is calculated based on the imact of cell size, sarsity of the solved matrix and number of oerations inside the cell. In this case the designer has three choices to control the cell size if the efficiency of the cell is too low.. CGE formula(eq. 3) is a comrehensive measure since it can be rewritten in four different forms that can be used to find: The efficiency of bloc-diagonal system and cell utilization at the same time by using only one formula (Eq. 6) since this equation contains the cell utilization measure (Eq. 8). The efficiency of bloc-diagonal system and cell indicator at the same time by using only one formula (Eq. 7) since this equation contains the cell indicator measure (Eq. 9). The efficiency of bloc-diagonal system and machine utilization at the same time by using only one formula (Eq. 9) since this equation contains the machine utilization measure (Eq. ). The efficiency of bloc-diagonal system and cell flexibility at the same time by using only one formula (Eq. ) since this equation contains the cell flexibility measure (Eq. 0) Derivation of cell utilization measure from CGE formula CGE measure can be rewritten as shown in Eq. (6). This formula contains the formula of cell utilization measure (Eq. 8). Eq. (6) can be used to find the efficiency of the main system and cell utilization at the same time as shown below in section v v v CGE... v e v e v e Let CU, CU and CU, where CU, CU and CU reresent the utilization of the first, the second and th cell, resectively. Let v, v and v. Then CGE can be rewritten as follows, CGE (6)... ( v e ) ( v e ) ( v e ) (5)

11 A. Muattash et al. / International Journal of Industrial Engineering Comutations 9 (08) Derivation of cell indicator measure from CGE formula CGE measure can be rewritten as shown below in Eq. (7). This formula contains the formula of cell indicator measure (Eq. 9). Eq. (7) can be used to find the efficiency of the main system, sarsity index and cell indicator at the same time as shown below in section CGE... ve v e v e v e v e and Let, v e. (7) where β, β and β are cell indicators of the first, the second and the th cell, resectively. In addition, α=/, α=/ and α=/ are sarsity index of the first, the second and the th cell, resectively. Therefore, we have CGE... (8) Derivation of machine utilization measure from CGE formula CGE measure can be rewritten as shown below in Eq. (9). This formula contains the formula of machine utilization measure (Eq. ). Eq. 9 can be used to find the efficiency of the main system and machine utilization at the same time as shown below in section P CGE.. v e v e v e.... Let MU P, where MU is the machine utilization. P Derivation of cell flexibility measure from CGE formula (9) CGE measure can be rewritten as shown below in Eq. (). This formula contains the formula of cell flexibility measure (Eq. (0)). Eq. () can be used to find the efficiency of the main system and cell flexibility at the same time as shown below in section P P CGE (0) e e P e P ( ) ( ) ( ) P P CGE... e e e () P P P Let MF, MF and MFP where MF, MF and MF reresent the flexibility of the first, second and th cell. Let, and then e e ep P CGE MF MF... MF. () Therefore the average measure of flexibility of the cells can be calculated as follows,

12 66 CGE AMF AMF AMF (3) Evaluation of the efficiency of bloc- diagonal system In this section, we assess the erformance of CGE through comaring the results with other well-nown measures mentioned in section. Eq. (3) will be used to find the efficiency of the system, sarsity index, efficacy index and efficiency of individual cells at the same time. For roblem X, Table 3 will show the details of the cell locations. Table 8 below summarizes the results. CGE In the same way we can find CGE for the other two roblems (Problem Y, Table 4 and Problem Z, Table 5). Table 8 Efficiency of the system and sub-system, Sarsity index, Efficacy index, using CGE for roblem Table roblem Efficiency of bloc -diagonal system (CGE) Sarsity Index Efficacy Index Efficiency of individual cells (sub-systems) Cell Cell Cell 3 Cell Cell Cell 3 Cell Cell Cell 3 3 X Y Z Consider the solution matrix of Table 8; it is clear that there is a consistency between these indicators. For examle in Table 8, the small values of sarsity index for cell and cell 3 means that these cells have small size, and in this case the designer can secify the uer and lower limit of machines to be allocated to each cell or to mae art assignment. Moreover, efficacy index and the efficiency of individual cells gave the same results regarding cell and cell 3. This means that cell has the highest efficiency, while cell and cell 3 has the lowest efficiency. As mentioned above any one of these three indicators (the efficiency of sub-system, sarsity index and efficacy index) will give the designer the oortunity to discover the cell that has the lowest efficiency in order to control the size. Without nowing the efficiency of sub-systems (individual cells), the system designer will not be able to control the cell size. Also the efficiency of bloc-diagonal system in this table (Table 8) using CGE grouing measure is very close to some well-nown measures in Table 6.The sueriority of CGE measure is that three cell indicators can be found concurrently with the efficiency of bloc-diagonal system Finding Cell Utilization using the CGE Formula Eq. (6) will be used to find the efficiency of the system and cell utilization at the same time. Problem X Table 3 will be solved in details. We have CU = 0.58, CU =0.75 and CU3=0.75. () (4) (4) GE (7 50) 4 0(30) 4 0(30) In the same way, the efficiency of the system and cell utilization for roblem Y, Table 4 and roblem Z, Table 5 can be found. The results are summarized below in Table 9. Table 9 Efficiency of the system and Cell utilization using CGE for roblem Table Problem Cell Utilization (CU) CGE CU CU CU 3 3 X Y Z

13 A. Muattash et al. / International Journal of Industrial Engineering Comutations 9 (08) 67 It is clear that cell utilization is the same in Table 9 and Table 7. CGE is the same as in Table 8. It is clear that instead of using two formulas to find the efficiency of the system and cell utilization, we use only one formula. In this case the designer will discover which cell has the lowest utilization. While the efficiency of the system will not give him any idea about the individual cells Finding Cell Indicator using the CGE Formula Eq. (7) will be used to find the efficiency of the system and cell indicator of the cells at the same time. Problem X Table 3 will be solved in details. 4 4 CGE α = 0.6, α = 0., α 3 = In the same way, the efficiency of the system and cell indicator for roblem Y Table 4 and roblem Z Table 5 can be found. The results are summarized below in Table 0. Table 0 Efficiency of the system and Cell indicator using CGE for roblem Table Problem CGE Cell Indicator (β) CU CU CU3 3 X Y Z It is clear that cell indicator is the same in Table 0 and Table 7 also CGE is the same as in Table 8. Knowing cell indicator will enable the designer to distinguish between ill-structured cell and erfectstructured cell. Moreover Eq. (7) can rovide the designer with sarsity index for every cell Finding Machine Utilization using the CGE Formula Eq. (9) will be used to find the efficiency of the system, machine utilization of the cells at the same time. Problem X Table 3 will be solved in details. The results are summarized below in Table CGE , which means the machine utilization is equal to In the same way the efficiency of the system and machine utilization for roblem Y Table 4 and roblem Z Table 5 can be found. Table Efficiency of the system and machine utilization using CGE for roblem Table Problem CGE Machine Utilization(MU) 3 X Y Z Using Eq. (9) will give the designer the same result as shown in Table 7 and Table 8.The designer can comare between the efficiency of the system and the machine utilization Finding Cell Flexibility using the CGE Formula Eq. () will be used to find the efficiency of the system, cell flexibility at the same time. Problem X Table 3 will be solved in details CGE 0.65, MF 0.35, 0.5 and 0.5 MF MF

14 68 AMF %, AMF % and AMF % In the same way the efficiency of the system and cell flexibility for roblem Y Table 4 and roblem Z Table 5 can be found. The results are summarized below in Table. Table Efficiency of the system and average measure of flexibility using CGE for roblem. Average Measure of Flexibility (AMF) Table Problem CGE Cell Cell Cell3 3 X % 5% 5% 4 Y % 9.8% 6.% 5 Z % 9.06% 4.33% From Table, the designer can find the efficiency of the system and cell flexibility at the same time by using only one formula (Eq. ) and the results are the same as in Table 7 and Table 8. Moreover, the designer can find the average measure of flexibility. 3.4 Comarison with Some Commonly Known Grouing Efficiency Measures In order to evaluate the erformance of the roosed measure, a comarison is erformed with different case studies taen from literature and the results are given in Table 3. CGE measure is comared with six well nown measures to evaluate the erformace of case studies. The result of CGE measure is found to be close to some of these grouing measures. It is clear that CGE measure is sensitive to number of voids, excetions, number of oerations inside the cell, sarsity of the system and sarsity of individual cells regardless of the size of the matrix. The sueriority of CGE measure over these measures is that, the designer can find the efficiency of individual cells, sarsity index and efficacy index for these cases which will give him the oortunity to control cell size. Moreover, since CGE measure can be rewritten with four different forms, then cell utilization and/or machine utilization and/or cell indicator and/or cell flexibility can be found to these cases. While the other measures can find only the efficiency of the system. 4. Conclusion Some well-nown grouing measures were discussed and analyzed. These erformance measures have some drawbacs such as ignoring sarsity of individual cells in the solved matrix and none of these measures can evaluate the efficiency of bloc-diagonal system and sub-system at the same time. Moreover, none of these measures can be used as a comrehensive grouing measure. In this study, to overcome the limitations of these measures, a new measure called Comrehensive Grouing Efficacy (CGE) was resented. The aroach adds the cell sarsity to its other elements in which the other measures ignore this issue, for that the efficiency of bloc-diagonal system and sub-system can be determined concurrently. CGE showed a better understanding of individual cells in the solved matrix, since the three cell indicators (sarsity index, efficacy index and efficiency of individual cells in the solved matrix) will hel the designer control the cell size. CGE is a comrehensive grouing measure since it can be used to find the efficiency of bloc-diagonal system and/or cell utilization and/or machine utilization and/or cell indicator and/or cell flexibility at the same time.

15 A. Muattash et al. / International Journal of Industrial Engineering Comutations 9 (08) 69 Table 3 Comarison of CGE with some commonly nown measures, (q=0.5) Problem Source urbidge (973), solved by octor (99) Problem size (n m) 6 43 # of cells () 4 Total number of oerations in the MP matrix 6 # of excetions (e) 6 # of voids (v) 77 Sarsit y () 77 Grouing efficacy (ω) CGE Grouing Efficiency (η) Weighted grouing efficacy (ω) Grouing Index (ϒ) Machine Utilization (MU) Modified grouing efficacy (τ) Yasuda and Yin (00) Muattash et al. (007) Pachayaan and Panneerselvam(05) Chen and Guerrero (994). Muattash et al. (0) Solved by Durga Rajesh et al. (06) o

16 70 Acnowledgement The authors would lie to than the anonymous referees for constructive comments on earlier version of this aer. References Al-ashir, A. A., Muattash, A. M., Muqattash, R. S., Al-Tal, S. Y., & Qamar, A. M. (06). Grouing Cell Indicator: A Modified Cell Formation Grouing Measure. Middle-East Journal of Scientific Research, 4(7), Asin, R. G., & Chiu, K. S. (990). A grah artitioning rocedure for machine assignment and cell formation in grou technology. International Journal of roduction Research, 8(8), Agrawal, A. K., hardwaj, P., & Srivastava, V. (0). On some measures for grouing efficiency. The International Journal of Advanced Manufacturing Technology, 56(5-8), octor, F.F. (996). The minimum-cost, machine-art cell formation roblem. International Journal of Production Research, 34(4), octor, F. F. (99). A linear formulation of the machine-art cell formation roblem. International Journal of Production Research, 9(), ottani, E., Centobelli, P., Cerchione, R., Gaudio, L., & Murino, T. (07). Solving machine loading roblem of flexible manufacturing systems using a modified discrete firefly algorithm. International Journal of Industrial Engineering Comutations, 8(3), urbidge, J. L. (973), Production flow analysis on a comuter. Third Annual Conference of the Institute of Production Engineers. ychov, I., atsyn, M., & Pardalos, P. M. (04). Exact model for the cell formation roblem. Otimization Letters, 8(8), rusco, M.J. (05), An exact algorithm for maximizing grouing efficacy in Part machine clustering. IIE Transactions, 47(6), Chandraseharan, M. P., & Rajagoalan, R. (986 a) An ideal seed non-hierarchical clustering algorithm for cellular manufacturing. International Journal of Production Research, 4(), Chandraseharan, M., & Rajagoalan, R. (986). MODROC: an extension of ran order clustering for grou technology. International Journal of Production Research, 4(5), -33. Chen, H. G., & Guerrero, H. H. (994). A general search algorithm for cell formation in grou technology. International Journal of Production Research, 3(), Elbenani,., & Ferland, J. A. (0). Cell formation roblem solved exactly with the dinelbach algorithm, Montreal. Quebec. CIRRELT-0-07,. 4 Harhalais, G., Nagi, R., & Proth, J. M. (990). An efficient heuristic in manufacturing cell formation for grou technology alications. The International Journal of Production Research, 8(), Ghosh, T.,., Doloi, P. K., Dan. (04). A novel cell formation technique in cellular manufacturing system based on roduction factors, 5 th International & 6th All India Manufacturing Technology, Design and Research Conference (AIMTDR 04), IIT Guwahati, Assam, India. Hsu, C.P. (990). Similarity coefficient aroaches to machine-comonent cell formation in cellular manufacturing: a comarative study. PhD thesis, Industrial and Systems Engineering, University of Wisconsin- Milwauee. Keeling, K.., rown, E. C., & James, T. L. (007). Grouing efficiency measures and their imact on factory measures for the machine-art cell formation roblem: A simulation study. Engineering Alications of Artificial Intelligence, 0(), Lee, K., & Ahn, K. I. (03). GT efficacy: a erformance measure for cell formation with sequence data. International Journal of Production Research, 5(0), Kusia, A., & Chow, W. S. (987). Efficient solving of the grou technology roblem. Journal of manufacturing systems, 6(), 7-4.

17 A. Muattash et al. / International Journal of Industrial Engineering Comutations 9 (08) 7 Mahdavi, I., Javadi,., Fallah-Aliour, K., & Slom, J. (007). Designing a new mathematical model for cellular manufacturing system based on cell utilization. Alied Mathematics and Comutation, 90(), Mansouri, S. A., Husseini, S. M., & Newman, S. T. (000). A review of the modern aroaches to multicriteria cell design. International Journal of Production Research, 38(5), 0-8. Muattash, A. M., Abbasi, G. Y., Tahboub, K. K., & Adil, M.. (006). A modified revised P-median aroach to cell formation. International Journal of Industrial and Systems Engineering, (), Muattash, A. M., Tahboub, K. K., Fouad, R. H., & Al-ashir, A. A. (0). A unified aroach for designing a cellular manufacturing system by secifying number of cells. International Journal of Alied Mathematics and Statistics, 7(3). Miltenburg, J., & Zhang, W. (99). A comarative evaluation of nine well-nown algorithms for solving the cell formation roblem in grou technology. Journal of Oerations Management, 0(), Murugan, M., & Selladurai, V. (0). Formation of machine cells/art families in cellular manufacturing systems using an ART-modified single linage clustering aroach a comarative study. Jordan Journal of Mechanical and Industrial Engineering, 5(3), 99-. Nagendra Parashar,.S. (004c), Evaluation of cellular manufacturing systems design- VEDO Analysis, Industrial Engineering Journal, 33(6), June,.4-8. Nair, G. J. K., & Narendran, T. T. (996). Grouing index: a new quantitative criterion for goodness of bloc-diagonal forms in grou technology. International Journal of Production Research, 34(0), Ng, S. M. (993). Worst-case analysis of an algorithm for cellular manufacturing. Euroean Journal of Oerational Research, 69(3), Pachayaan, M., & Panneerselvam, R. (05). Hybrid Genetic Algorithm for Machine-Comonent Cell Formation. Intelligent Information Management, 7(03), 07. Rabbani, M., Elahi, S., & Javadi,. (07). A comrehensive quadratic assignment roblem for an integrated layout design of final assembly line and manufacturing feeder cells. Decision Science Letters, 6(), Rajesh, K. D., Chalaathi, P. V., Chaitanya, A.. K., Sairam, V., & Anildee, N. (006). Modified grouing efficacy and new average measure of flexibility: erformance measuring arameters for cell formation alications, ARPN Journal of Engineering and Alied Sciences, (5), Rezazadeh, H., & Khiali-Miab, A. (07). A two-layer genetic algorithm for the design of reliable cellular manufacturing systems. International Journal of Industrial Engineering Comutations, 8(3), Sarer,. R. (999). Grouing efficiency measures in cellular manufacturing: a survey and critical review. International Journal of Production Research, 37(), Sarer,. R. (00). Measures of grouing efficiency in cellular manufacturing systems. Euroean Journal of Oerational Research, 30(3), Sarer,. R., & Khan, M. (00). A comarison of existing grouing efficiency measures and a new weighted grouing efficiency measure. Iie Transactions, 33(), -7. Seifoddini, H., & Djassemi, M. (996). A new grouing measure for evaluation of machine-comonent matrices. International Journal of Production Research, 34(5), Suresh Kumar, C., & Chandraseharan, M. P. (990). Grouing efficacy: a quantitative criterion for goodness of bloc diagonal forms of binary matrices in grou technology. International Journal of Production Research, 8(), Wu, N. (998). A concurrent aroach to cell formation and assignment of identical machines in grou technology. International Journal of Production Research, 36(8), Yasuda, K., & Yin, Y. (00). A dissimilarity measure for solving the cell formation roblem in cellular manufacturing. Comuters & Industrial Engineering, 39(), -7.

18 7 07 by the authors; licensee Growing Science, Canada. This is an oen access article distributed under the terms and conditions of the Creative Commons Attribution (CC- Y) license (htt://creativecommons.org/licenses/by/4.0/).